BEING CERTAIN WITH UNCERTAINTY

How they measure chance? what are the basic operations in probability? Let us find out.
In tossing of a coin, there are two events-- head and tail. The chance one event occurring = one/total number of events. That is 1/2. You have 50% chance of getting head or tail for each toss. If you toss a coin, one time you should get head and next time you should get tail, so the formula says. But both the times, you may get head or tail. If you toss the coin 100 times, roughly 50 times you will get head and tail. On the long run, the formula proves correct.
On throwing dice, there are six events (1,2,3,4,5,6). Hence chance of getting an 'event' for a throw is 1/6. There are three important logical operations in probability. Let us go for it.

OR operation:
What is the chance or probability getting 1 or 6 in a roll. Here, either 1 or 6 will do for us.
Probability of getting 1 is 1/6
probability of getting 6 is 1/6
Now add both
1/6 + 1/6 =2/6 =1/3
1/3 is the chance of getting 1 or 6. Hence for 'OR' operation, add the probabilities.

AND operation:
What is the chance of getting 1 AND 1 consecutively in two rolls. We know, probability for each event is 1/6.
For AND operation, multiply the probabilities.
(1/6) * (1/6) = 1/36
1/36 is the probability of getting a one AND a one, one after the other. 1/36 is a very small number. Hence the occurrence of the event is rare and unlikely.

NOT operation:
What is the probability of NOT rolling a six. That is, we may get 1,2,3,4,5 but not 6. The total probability is always '1'. The probability of rolling 6 is 1/6. Now subtract 1/6 from 1
1-1/6 = 5/6
5/6 is the probability of rolling 1 or 2 or 3 or 4 or 5 and NOT 6. 5/6 is large value. Hence 'Not throwing 6' is very likely to happen.

By combining the three logic operations in various ways, we can answer many questions. For example: if you are dealt five random cards from standard deck of 52, what are the chance of getting one pair whose values match(a pair queen or a pair of A ) and no other matching cards? It is left to the reader as a home work!

FOOT NOTE: Using the same three logic operations and binary numbers, computer is able to do miracles. So does the probability theory.
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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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